The Funk-Hecke formula, harmonic polynomials, and derivatives of radial distributions

Abstract

We give a version of the Funk-Hecke formula that holds with minimal assumptons
and apply it to obtain formulas for the distributional derivatives of radial
distributions in Rn of the type
Yk
􀀀
r


j
(f (r)) ;
where Yk is a harmonic homogeneous polynomial. We show that such derivatives have
simpler expressions than those of the form p
􀀀
r

(f (r)) for a general polynomial p: